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Unformatted text preview: 1 Homework 2
U = (x, y, z, w) R4 x - y - z = 0, z - w = 0 , W = (x, y, z, w) R4 y - z + w = 0, x - z = 0 , V = (x, y, z, w) R4 - x + y + w = 0 . 1. Let U, V, W R4 be the subspaces a. Prove that V = U + W . Proving that U + W V is easier, but to show that V U + W , you may have to solve some equations, much like in class. b. Is the sum direct? How do you know? 2. Let U, W, V, Uk , Wk , Vk be as in the symmetric function example from class, and the notes in the "documents" section on blackboard. We showed that Vk = Uk Wk . Assuming this, prove that V is the direct sum of the infinite-dimensional vector spaces V = U W , just by using the definitions. 3. If U, U are subspaces of V , then the union U U is almost never a subspace (unless one happens to be contained in the other). Prove that, if W is a subspace, and U U W , then U + U W . 4. Axler, chapter 2, # 2,8,9,12. (hint, use proposition 2.17 for number 12). 1 ...
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